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A031439
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a(0) = 1, a(n) is the greatest prime factor of a(n-1)^2+1 for n > 0.
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7
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1, 2, 5, 13, 17, 29, 421, 401, 53, 281, 3037, 70949, 1713329, 1467748131121, 37142837524296348426149, 101591133424866642486477019709, 1650979973845742266714536305651329, 78343914631785958284737, 4029445531112797145738746391569, 350080544438648120162733678636001, 26208090024628793745288451837610346882122253572537, 4717815978577117335515270825550279551117660519482308365269206484133871485221
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OFFSET
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0,2
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COMMENTS
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All a(n) except a(0) = 1 belong to A014442(n) = {2, 5, 5, 17, 13, 37, 5, 13, 41, 101, ...} Largest prime factor of n^2 + 1. All a(n) except a(0) = 1 belong to A002313(n) = {2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, ...} Primes congruent to 1 or 2 modulo 4; or, primes of form x^2+y^2; or, -1 is a square mod p. All a(n) except a(0) = 1 and a(1) = 2 are the Pythagorean primes A002144(n) = {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, ...} Primes of form 4n+1. - Alexander Adamchuk, Nov 05 2006
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LINKS
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EXAMPLE
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A006530(101591133424866642486477019709^2+1)=
A006530(10320758390549056348725939119133160378521185060950774444682)=
A006530(2*29*23201*4645528280970018601*1650979973845742266714536305651329)=
1650979973845742266714536305651329, factorization of A006530(a(15)^2+1) by Dario A. Alpern's program (see link).
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MATHEMATICA
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gpf[n_] := FactorInteger[n][[-1, 1]]; a[0] = 1; a[n_] := a[n] = gpf[a[n - 1]^2 + 1]; Table[an = a[n]; Print[an]; an, {n, 0, 21}] (* Jean-François Alcover, Nov 04 2011 *)
NestList[FactorInteger[#^2+1][[-1, 1]]&, 1, 21] (* Harvey P. Dale, Jul 04 2013 *)
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PROG
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(PARI) gpf(n)=local(pf); pf=factor(n); pf[matsize(pf)[1], 1] vector(20, i, r=if(i==1, 1, gpf(r^2+1)))
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CROSSREFS
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Cf. A002144 - Pythagorean primes: primes of form 4n+1; A002313 - Primes congruent to 1 or 2 modulo 4; A014442 - Largest prime factor of n^2 + 1.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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a(17)-a(21) from Richard FitzHugh (fitzhughrichard(AT)hotmail.com), Aug 12 2004
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STATUS
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approved
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